where only a finite number of decompositions contribute (this follows from
\(\Im (e^{-I\psi }Z_\sigma (\gamma _i))=|Z(\gamma _i)|\)  and the support condition).
The coefficients \(g(\lbrace \gamma _i\rbrace )\)  can be computed either using
the attractor tree formula of [1]} or the flow tree formula of [2]}The Coulomb branch formula of [3]}, [4]} gives yet another prescription, which gives
the same coefficient \(g(\lbrace \gamma _i\rbrace )\)  whenever
all charges \(\gamma _i\)  whose rays intersect \(\sigma \)  lie in a common two-dimensional sublattice, such that scaling solutions do not occur..
Both involve a sum over rooted trees decorated with
charges \(\gamma _e\)  along the edges and stability parameters \(\theta _v\)  at the vertices,
valued in an auxiliary space \({R}^n\) . The former involves trees of arbitrary valency
and stability parameters at the root vertex
given by \(\theta _i=\Re (e^{-I\psi } Z_\sigma (\gamma _i))\) , while the latter involves binary
trees only, at the cost of perturbing the stability parameters at the root vertex.
